Fashion-Design degree here... Create nice finished seams around the edge of each of your designs to make them into patches. Trim off the excess fabric. Re-make your bunny's body parts. Apply the patches to the finished bunny's body parts. You simply don't have enough seam allowance around the edge. Oh. And get some fray-check. It is essentially sewing glue that you put on fraying edges like that, that helps keep them from coming apart. Dab on the fray-check, fold over the edge, sew, iron, and it will last forever.
If you don't mind, can I ask a material question? Is my twill fraying so much because it's threads are thicker or because the weave and weft threads are an uneven ratio? None of the cotton I use ever frays this much, but I don't know if that has to do with thread size or because that fabric tends to have an even weave and weft ratio. I do know that the embroidery is a bit more difficult because the uneven thread ratio makes it impossible for me to count threads, but a) I'm kind of just freestyling and b) my ehes aren't what they used to be so I don't even know if I'd want to go back to counting threads.
Embroidery is not something I have done. But it seems like you need to have a 'well-made' fabric if you are counting warp and weft threads... if the threads are your grid, then loosely-woven fabric, small pieces that can be easily racked or twisted, or fabric with large threads (canvas-like cottons), could all be an issue. It would seem that embroidery should be done on a piece of fabric with finished edges. Sew a basic rolled scarf seam around the edge, and that will ensure the threads don't slip and the fabric doesn't warp on the bias, and throw your grid off. Then, when done, cut off the seam by trimming the embroidered piece to shape/size. So, in short, to answer your question, I'd say both things are an issue with this specific piece. And the problem made worse because the size of the pieces were rather small, had no seam allowance, and working the piece with the embroidery needles just caused the threads to loosen and fray with too much handling. Does that help?